MinT - Best Known (166−33, 166, s)-Nets in Base 2

Best Known (166−33, 166, s)-Nets in Base 2

(166−33, 166, 260)-Net over F₂ — Constructive and digital

Digital (133, 166, 260)-net over F₂, using

t-expansion [i] based on digital (132, 166, 260)-net over F₂, using
- 2 times m-reduction [i] based on digital (132, 168, 260)-net over F₂, using
  - trace code for nets [i] based on digital (6, 42, 65)-net over F₁₆, using
    - net from sequence [i] based on digital (6, 64)-sequence over F₁₆, using
      - Niederreiterâ€“Xing sequence constructionÂ II/III [i] based on function field F/F₁₆ with g(F) = 6 and N(F) ≥ 65, using
        the Hermitian function field over F₁₆ [i]

(166−33, 166, 491)-Net over F₂ — Digital

Digital (133, 166, 491)-net over F₂, using

embedding of OOA with Gilbertâ€“VarÅ¡amov bound [i] based on linear OOA(2¹⁶⁶, 491, F₂, 2, 33) (dual of [(491, 2), 816, 34]-NRT-code), using
- discarding factors / shortening the dual code based on linear OOA(2¹⁶⁶, 524, F₂, 2, 33) (dual of [(524, 2), 882, 34]-NRT-code), using
  - 2¹ times duplication [i] based on linear OOA(2¹⁶⁵, 524, F₂, 2, 33) (dual of [(524, 2), 883, 34]-NRT-code), using
    - OOA 2-folding [i] based on linear OA(2¹⁶⁵, 1048, F₂, 33) (dual of [1048, 883, 34]-code), using
      - 1 times code embedding in larger space [i] based on linear OA(2¹⁶⁴, 1047, F₂, 33) (dual of [1047, 883, 34]-code), using
        adding a parity check bit [i] based on linear OA(2¹⁶³, 1046, F₂, 32) (dual of [1046, 883, 33]-code), using
        constructionÂ XX applied to C₁Â =Â C([1021,28]), C₂Â =Â C([1,30]), C₃Â =Â C₁Â +Â C₂Â =Â C([1,28]), and C_âˆ©Â =Â C₁Â âˆ©Â C₂Â =Â C([1021,30]) [i] based on
        linear OA(2¹⁵¹, 1023, F₂, 31) (dual of [1023, 872, 32]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,28}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 32 [i]
        linear OA(2¹⁵⁰, 1023, F₂, 30) (dual of [1023, 873, 31]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,30], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 31 [i]
        linear OA(2¹⁶¹, 1023, F₂, 33) (dual of [1023, 862, 34]-code), using the primitive BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â {−2,−1,…,30}, and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 34 [i]
        linear OA(2¹⁴⁰, 1023, F₂, 28) (dual of [1023, 883, 29]-code), using the primitive narrow-sense BCH-code C(I) with length 1023Â = 2¹⁰âˆ’1, defining interval IÂ =Â [1,28], and designed minimum distance dÂ â‰¥Â |I|+1Â =Â 29 [i]
        linear OA(2¹, 12, F₂, 1) (dual of [12, 11, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s, using
        parity-check code [i]
        linear OA(2¹, 11, F₂, 1) (dual of [11, 10, 2]-code), using
        discarding factors / shortening the dual code based on linear OA(2¹, s, F₂, 1) (dual of [s, s−1, 2]-code) with arbitrarily large s (see above)

(166−33, 166, 8624)-Net in Base 2 — Upper bound on s

There is no (133, 166, 8625)-net in base 2, because

1 times m-reduction [i] would yield (133, 165, 8625)-net in base 2, but
- the generalized Rao bound for nets shows that 2^mÂ â‰¥ 46 817669 846027 915524 728347 208751 617486 537961 916151 > 2¹⁶⁵ [i]

Best Known (166−33, 166, s)-Nets in Base 2

(166−33, 166, 260)-Net over F2 — Constructive and digital

(166−33, 166, 491)-Net over F2 — Digital

(166−33, 166, 8624)-Net in Base 2 — Upper bound on s

(166−33, 166, 260)-Net over F₂ — Constructive and digital

(166−33, 166, 491)-Net over F₂ — Digital